4
Convolution
In Lecture 3 we introduced and defined a variety of system properties to
which we will make frequent reference throughout the course. Of particular
importance are the properties of linearity and time invariance, both because
systems with these properties represent a very broad and useful class and because with just these two properties it is possible to develop some extremely
powerful tools for system analysis and design.
A linear system has the property that the response to a linear combination of inputs is the same linear combination of the individual responses. The
property of time invariance states that, in effect, the system is not sensitive to
the time origin. More specifically, if the input is shifted in time by some
amount, then the output is simply shifted by the same amount.
The importance of linearity derives from the basic notion that for a linear
system if the system inputs can be decomposed as a linear combination of
some basic inputs and the system response is known for each of the basic inputs, then the response can be constructed as the same linear combination of
the responses to each of the basic inputs. Signals (or functions) can be decomposed as a linear combination of basic signals in a wide variety of ways. For
example, we might consider a Taylor series expansion that expresses a function in polynomial form. However, in the context of our treatment of signals
and systems, it is particularly important to choose the basic signals in the expansion so that in some sense the response is easy to compute. For systems
that are both linear and time-invariant, there are two particularly useful
choices for these basic signals: delayed impulses and complex exponentials.
In this lecture we develop in detail the representation of both continuoustime and discrete-time signals as a linear combination of delayed impulses
and the consequences for representing linear, time-invariant systems. The resulting representation is referred to as convolution. Later in this series of lectures we develop in detail the decomposition of signals as linear combinations of complex exponentials (referred to as Fourieranalysis) and the
consequence of that representation for linear, time-invariant systems.
In developing convolution in this lecture we begin with the representation of discrete-time signals and linear combinations of delayed impulses. As
we discuss, since arbitrary sequences can be expressed as linear combinations of delayed impulses, the output for linear, time-invariant systems can be
Signals and Systems
4-2
expressed as the same linear combination of the system response to a delayed
impulse. Specifically, because of time invariance, once the response to one
impulse at any time position is known, then the response to an impulse at any
other arbitrary time position is also known.
In developing convolution for continuous time, the procedure is much
the same as in discrete time although in the continuous-time case the signal is
represented first as a linear combination of narrow rectangles (basically a
staircase approximation to the time function). As the width of these rectangles becomes infinitesimally small, they behave like impulses. The superposition of these rectangles to form the original time function in its limiting form
becomes an integral, and the representation of the output of a linear, time-invariant system as a linear combination of delayed impulse responses also becomes an integral. The resulting integral is referred to as the convolution integral and is similar in its properties to the convolution sum for discrete-time
signals and systems. A number of the important properties of convolution that
have interpretations and consequences for linear, time-invariant systems are
developed in Lecture 5. In the current lecture, we focus on some examples of
the evaluation of the convolution sum and the convolution integral.
Suggested Reading
Section 3.0, Introduction, pages 69-70
Section 3.1, The Representation of Signals in Terms of Impulses, pages 70-75
Section 3.2, Discrete-Time LTI Systems: The Convolution Sum, pages 75-84
Section 3.3, Continuous-Time LTI Systems: The Convolution Integral, pages
88 to mid-90
Convolution
MARKERBOARD
4.1
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TRANSPARENCY
4.1
A general discretetime signal expressed
as a superposition of
weighted, delayed unit
impulses.
Signals and Systems
TRANSPARENCY
4.2
jj)x[
The convolution sum
for linear, timeinvariant discrete-time
systems expressing
the system output as a
weighted sum of
delayed unit impulse
responses.
x[2]
-1 0
I
2
X10{
-I
x[0]8[n]
-0-0-0-0--
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0
m-
x[0] h [n]
n
x
X x[1]8[n+1]
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n
x[-2]8[n+2]
x [-I] h [n+1.
x [-2] h [n.2]
-1 0 i 2
0
x[k] S[n-k]
x[n] =E
k = -o
TRANSPARENCY
4.3
One interpretation of
the convolution sum
for an LTI system.
Each individual
sequence value can be
viewed as triggering a
response; all the
responses are added
to form the total
output.
[ 1] h [n-I]
0
Linear System:
+0n
y [n] =E
x[k] hk [n]
k = - 010
5 [n - k] - h [n]
If Time-invariant:
hk [n] = h [n-k]
LTI: y[n]
+
o0
=E
x[k] h[n - k]I
Convolution Sum
Convolution
4-5
x(-2A)&a(t+2A)A
x(-2A)
-2A
t
-A
-A 0
t
X (0)
I X(0)
0
BA
TRANSPARENCY
4.4
Approximation of a
continuous-time signal
as a linear
combination of
weighted, delayed,
rectangular pulses.
[The amplitude of the
fourth graph has been
corrected to read
x(O).]
A
t
A
t
- xA)
x (A)
A
86
(t-A)
t
2A
x(o) 6A(t) A + x(A) 6 A(t
x(t)
+ x(- A) 6A(t + A)A+...
TRANSPARENCY
4.5
As the rectangular
pulses in Trans-
X(t)
x(t)
x(k A) 6,(t - k A) A
=
lim 1
A+O
x(k A) S(t - k A) A
k=-oc
+W
x(,r) 6(t -,r) d-r
f --00
parency 4.4 become
increasingly narrow,
the representation
approaches an
integral, often referred
to as the sifting
integral.
Signals and Systems
4-6
x(t)
(
Eim
x(k A) 56(t - kA) A
=
'L+0 k=-o
TRANSPARENCY
4.6
Linear System:
Derivation of the
convolution integral
representation for
continuous-time LTI
systems.
y(t) =
0
+o
+O k=-
x(kA) hk(t)
A
o
+00
=f
xT) hT(t) dr
If Time-Invariant:
hkj t) = ho(t - kA)
= he (t - r)
h,(t)
+01
LTI:
v(t)
f x(r) h(t-7) dr-0
1
Convolution Integral
x(t)
t
0
ti
TRANSPARENCY
4.7
Interpretation of the
convolution integral as
a superposition of the
responses from each
of the rectangular
pulses in the
representation of the
input.
x (0) h Mt
x (0)
x(A)
x(kA)
oA
kA
AA
y(t)
0
t
x(t)
0
y(t)
t
oA
t
Convolution
4-7
Convolution Sum:
+0o
x[n] =E x[k] S[n-k]
k= -0ok
y [n]
=
x [k]
k= -o00
h[n-k] =x[n]
*
h[n]
TRANSPARENCY
4.8
Comparison of the
convolution sum for
discrete-time LTI
systems and the
convolution integral
for continuous-time
LTI systems.
Convolution Integral:
+00
x(t)
=f
x(-) 6(t-r)
dr
+fd
y(t)
=
X(r) h(t-,r) dr-= x(t) -*h(t)
y [n]
Z
x[k]h
[n-k]
x [n]= u [n]
TRANSPARENCY
4.9
h[n]=an u[n]
x [n]
0
n
h [n]
x [k]
t
k
h [n-k]
n
k
Evaluation of the
convolution sum for
an input that is a unit
step and a system
impulse response that
is a decaying
exponential for n > 0.
Signals and Systems
y(t)f
x(t)
TRANSPARENCY
4.10
Evaluation of the
convolution integral
for an input that is a
unit step and a system
impulse response that
is a decaying
exponential for t > 0.
x(r)h(t-r)dr
u (t)
h (t )=e~43
u t)
x (t)
t
O
r
0
h (t)
x (r)
h (t-r)
t
MARKERBOARD
4.2
T
Convolution
MARKERBOARD
4.3
v4egva.z:
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Lk (t- -CjT
t).
t
k Lt-T)aT
Ct
1
U -e
(0
3
oov0
eoverkp ~3et ee
C-t
t
h*O
L
t,<0o
E
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r)
t
t
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Signals and Systems
Professor Alan V. Oppenheim
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